01_handout_1(3).pdf

SH1657

Definition 1.1. lim #($) = % if and only if ∀ * > 0, ∃ a real , > 0 such that whenever 0 < |$ − .| < ,, then →"

|#($) − %| < *. Definition 1.2 We say that % is the left-hand limit of #($) at ., written lim/ #($) = % →"

if ∀ * > 0, ∃ a real , > 0 such that whenever . − , < $ < ., |#($) − %| < *. We say that % is the right-hand limit of #($) at ., written lim1 #($) = % →"

if ∀ * > 0, ∃ a real , > 0 such that whenever . < $ < . + ,, |#($) − %| < *. Definition 1.3 If a function # increases or decreases without bound as $ approaches a real number . from either the right or the left, then # has a vertical asymptote at $ = .. A function has a vertical asymptote at $ = . if one of the following conditions are satisfied: a.

lim1 #($) = ∞ and lim/ #($) = ∞ →"

→"

b. lim1 #($) = ∞ and lim/ #($) = −∞ →"

c.

→"

lim1 #($) = −∞ and lim/ #($) = ∞ →"

→"

d. lim1 #($) = −∞ and lim/ #($) = −∞ →"

→"

Definition 1.4 If the values of a function #($) approach a real number % as $ increases or decreases without bound, then # has a horizontal asymptote at 4 = %. A non-constant function has a horizontal asymptote at 4 = % if one of the following conditions are satisfied: a. lim #($) = % →5

b.

lim/ #($) = % →5

References: Azad, K. Better Explained. (n.d.). AA Gentle Introduction to Learning Calculus. Retrieved from: http://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/ last May 18, 2016. Coburn, J. (2016). Pre-Calculus. McGraw Hill Education. Minton, R. & Smith, R. (2016). Basic Calculus. McGraw Hill Education. 01 Handout 1

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